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arXiv:1409.2256 · Physical Review A

Realization of fractional Chern insulators in the thin-torus limit with ultracold bosons

Fabian Grusdt, Michael Höning

Published Nov 19, 2014• 68 citations
arXivHTMLPDF

Abstract

Topological states of interacting many-body systems are at the focus of current research due to the exotic properties of their elementary excitations. In this paper we suggest a realistic experimental setup for the realization of a simple version of such a phase. We show how $\ensuremath{\delta}$-interacting bosons hopping on the links of a one-dimensional ladder can be used to simulate the thin-torus limit of the two-dimensional (2D) Hofstadter-Hubbard model at one-quarter magnetic flux per plaquette. Bosons can be confined to ladders by optical superlattices, and synthetic magnetic fields can be realized by laser-assisted tunneling. We show that twisted boundary conditions can be implemented, enabling the realization of a fractionally quantized Thouless pump. Using numerical density-matrix-renormalization-group calculations, we show that the ground state of our model is an incompressible symmetry-protected topological charge density wave phase at average filling $\ensuremath{\rho}=1/8$ per lattice site, related to the $1/2$ Laughlin-type state of the corresponding 2D model.

DOI
10.1103/physreva.90.053623

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Topological states of interacting many-body systems are at the focus of current research due to the exotic properties of their elementary excitations. In this paper we suggest a realistic experimental setup for the realization of a simple version of such a phase. We show how \ensuremathδ\ensuremath{\delta}\ensuremathδ-interacting bosons hopping on the links of a one-dimensional ladder can be used to simulate the thin-torus limit of the two-dimensional (2D) Hofstadter-Hubbard model at one-quarter magnetic flux per plaquette. Bosons can be confined to ladders by optical superlattices, and synthetic magnetic fields can be realized by laser-assisted tunneling. We show that twisted boundary conditions can be implemented, enabling the realization of a fractionally quantized Thouless pump. Using numerical density-matrix-renormalization-group calculations, we show that the ground state of our model is an incompressible symmetry-protected topological charge density wave phase at average filling \ensuremathρ=1/8\ensuremath{\rho}=1/8\ensuremathρ=1/8 per lattice site, related to the 1/21/21/2 Laughlin-type state of the corresponding 2D model.
Topological order, the hallmark of fractional quantum Hall states, is primarily defined in terms of ground-state degeneracy on higher-genus manifolds, e.g., the torus. We investigate analytically and numerically the smooth crossover between this topological regime and the Tao-Thouless thin torus quasi-one-dimensional (1D) limit. Using the wire-construction approach, we analyze an emergent charge density wave (CDW) signifying the breakdown of topological order, and relate its phase shifts to Wilson loop operators. The CDW amplitude decreases exponentially with the torus circumference once it exceeds the transverse correlation length controllable by the interwire coupling. By means of numerical simulations based on the matrix product states (MPS) formalism, we explore the extreme quasi-1D limit in a two-leg flux ladder and present a simple recipe for probing fractional charge excitations in the $\ensuremath{\nu}=1/2$ Laughlin-like state of hard-core bosons. We discuss the possibility of realizing this construction in cold-atom experiments. We also address the implications of our findings to the possibility of producing non-Abelian zero modes. As known from rigorous no-go theorems, topological protection for exotic zero modes such as parafermions cannot exist in 1D fermionic systems and the associated degeneracy cannot be robust. Our theory of the 1D--2D crossover allows us to calculate the splitting of the degeneracy, which vanishes exponentially with the number of wires, similarly to the CDW amplitude.
We study the topological properties of an extended Bose-Hubbard model with cyclically modulated hopping and on-site potential parameters, which can be realized with ultracold bosonic atoms in a one-dimensional optical superlattice. We show that the interacting bosonic chain at half filling and in the deep Mott insulating regime can simulate bosonic Chern insulators with a topological phase diagram similar to that of the Haldane model of noninteracting fermions. Furthermore, we explore the topological properties of the ground state by calculating the many-body Chern number, the quasiparticle energy spectrum with gapless edge modes, the topological pumping of the interacting bosons, and the topological phase transition from normal (trivial) to topological Mott insulators. We also present the global phase diagram of the many-body ground state, which contains a superfluid phase and two Mott insulating phases with trivial (a zero Chern number) and nontrivial topologies (a nonzero Chern number), respectively.
The recent progress in engineering topological band structures in optical-lattice systems makes it promising to study fractional Chern insulator states in these systems. Here we consider a realistic finite system of a few repulsively interacting bosons on a square lattice with magnetic flux and sharp edges, as it can be realized in quantum-gas microscopes. We investigate under which conditions a fractional Chern insulator state corresponding to the Laughlin-like state at filling \nu=1/2 ν = 1 / 2 can be stabilized and its fractional excitations probed. Using numerical simulations, we find an incompressible bulk density at the expected filling for systems, whose linear extent is as small as 6-8 sites. This is a promising result, since such small systems are favorable with respect to the required adiabatic state preparation. Moreover, we also see very clear signatures of excitations with fractional charge in response both to static pinning potentials and dynamical flux insertion. Since the compressible edges, which are found to feature chiral currents, can serve as a reservoir, these observations are robust against changes in the total particle number. Our results suggest that signatures of both a fractional Chern insulator state and its fractional excitations can be found under realistic experimental conditions.
Synthetic ladders pierced by a magnetic field realized with one-dimensional alkaline-earth(-like) fermionic gases represent a promising environment for the investigation of quantum-Hall physics with ultracold atoms. We unveil the existence of a hierarchy of insulating and conducting states which appear at fractional filling factors by means of both analytical and numerical methods based on the density-matrix renormalization group algorithm. Analogously to the original quantum Hall effect, we show that such states can be exploited for constructing a topological Thouless pump where the charge transported after one cycle is quantized to fractional values and demonstrate this behavior with a full many-body time-dependent calculation. We conclude with some results on Laughlin-like states in both fermionic and bosonic systems.
We analytically and numerically analyze the one-dimensional "thin-torus" limit of Fractional Topological Insulators in a series of simple models exhibiting exactly flat bands with local hopping. These models are the one-dimensional limit of two dimensional Chern Insulators, and the Hubbard-type interactions projected into their lowest band take particularly simple forms. By exactly solving the many-body interacting spectrum of these models, we show that, just like in the Fractional Quantum Hall effect, the zero modes of the thin-torus limit are CDW states of occupation numbers satisfying generalized Pauli principles. As opposed to the FQH where the thin-torus CDW appear in orbital space, in the thin-torus FCI states, the CDW states are in real-space. We show the counting of the quasihole excitations in the energy spectrum cannot distinguish between a CDW state and a FQH state. However, by exactly computing the entanglement spectrum for the thin-torus states, we show that it can qualitatively and quantitatively distinguish between a CDW and a fractional topological state such as the FCI. We then discover a previously unknown separation of energy scales of the full FQH energy spectrum in the thin torus limit and find that Chern insulator models exhibiting strong isotropic FCI states have a similar structure in their thin-torus limit spectrum. We close by numerically computing the evolution of energy and entanglement spectra from the thin-torus to the isotropic limit. Our results can also be interpreted as an analysis of one-body, 1-dimensional topological insulators stabilized by inversion symmetry in the presence of interactions.
We present a setup with Rydberg atoms for the realization of a bosonic fractional Chern insulator in artificial matter. The suggested setup relies on Rydberg atoms arranged in a honeycomb lattice, where excitations hop through the lattice by dipolar exchange interactions and can be interpreted as hard-core bosons. The quantum many-body Hamiltonian is studied within exact diagonalization and the density-matrix renormalization group. We identify experimentally accessible parameters where all signatures indicate the appearance of a fractional state with the same topological properties as the ν=1/2 bosonic Laughlin state. We demonstrate an adiabatic ramping procedure, which allows for the preparation of the topological state in a finite system, and demonstrate an experimentally accessible smoking-gun signature for the fractional excitations.